Cosmic Rays and the Search for

a Violation of Lorentz Invariance

I. Cosmic raysPhenomenology, GZK cutoff

II. Violation of Lorentz InvarianceTheory, maximal attainable velocitiesApplications to decays, Cerenkov radiation,GZK cutoff and neutrino oscillation

III. Cosmic γ-raysRadiation of active galactic nuclei and γ-Ray-Bursts(Photons in a non-commutative space)

IV. Summary and Appendix News since Nov. 2007

I. Cosmic Rays

High energy particles and nuclei of cosmic origin

1912 Discovery by V.F. Hess with an electroscope on a balloon :ionising radiation decreases up to 2000 m (radioactivity from earth)

but rises again above [Hess: → 5300 m; W. Kolhorster (1913/4) → 9000 m]

⇒ Cosmic origin (beyond sun)

1938 P. Auger : separated Geiger counters detect correlated events.Extended Air Shower

Cascade triggered by a high energy primary particle. Estimate: E ≥ 1015 eV

Knowledge today: E ≈ 109 . . . 1020 eV

at ultra high energy : ≈ 90 % protons, 9 % α, plus a few heavier nuclei

Isotropic; for charged rays source cannot be located (traditional picture)

[ deflection by interstellar magnetic fields ∼ O(µG) ]

1

Flux of cosmic rays vs. energy, over broad interval essentially ∝ E−3

2

Origin ??? Historic proposal: Fermi mechanismsCollisions in a magnetic cloud.Later version : shock waves in gas of a supernova.Explanation at best up to ∼ 1014 eV.

Predicts energy density ∝ E−2 [ Observations close to E−3 ]

Two classes of scenarios :

• “Bottom-up” : Sources build up huge energy(at least 7 orders of magnitude beyond man-made accelerators)

Pulsars ? (rotating magnetised neutron stars) Quasars ? (→ Sect. 2)

Active Galactic Nuclei ? (→ Appendix)

• “Top-down” : Decay of super-heavy particlesgenerated in Big Bang → energy available(magnetic monopoles, “wimpzillas” . . . ??)

3

1965 A. Penzias/R. Wilson discover Cosmic Microwave Background

Relic of the Big Bang, photons decoupled after ≈ 3.8 · 105 years

Very precise Planck distribution : dnγ/dω ∝ω2

eω/kT−1, ω : γ-energy

0 5 10 15 20

n γ

ω [10-4 eV]

T = 2.725(1) K 〈ω〉 = 6 · 10−4 eV ,R ∞0 dω nγ

′(ω) ≃ 411 cm−3 〈λ〉 ≃ 1.9 mm (microwaves)

4

5

GZK Cutoff

1966 K. Greisen (Cornell), G.T. Zatsepin und V.A. Kuz’min (Lebedev)

Prediction: Cosmic rays have cutoff at E ≈ 6 · 1019 eV

Reason: Photopion production, in particular:

p + γ → ∆(1232 MeV)→ p + π0

→ n + π+ , n → p + e− + νe

}

99.4%

[ Further resonances: ∆(1620, 1700, . . . ) , p∗(1440, 1520 . . . ) etc.→ p + π or p + 2π ]

6

Threshold for proton energy: Ep = E0

s = (E0 + ω)2 − (~pp + ~pγ)

2(′′laboratory

′′, FRW metrics)

= E20 − ~p

2p

| {z }

m2p

+2E0 ω − 2~pp ~pγ ≃ m2p + 4E0 ω| {z }

head−on

!=m

2∆ (rest frame of ∆)

E0 =m2

∆ −m2p

4ω

e.g. ω=5〈ω〉z}|{= 6 · 1019

eV

Further kinemat. transformations → Inelasticity K :=∆EpEp

= 12

h

1 − m2p−m2

πs

i

Rest frame of the proton: s = (mp + ω)2 − ~p 2γ = m2

p + 2mp ω

Doppler effect: ω = γω(1− vpc cos θ) (θ : scattering angle in “laboratory”)

γ =Epmp

, e.g.E0mp

∼ 1011 ⇒ 〈ω〉θ ≃ 180 MeV · EpE0

K(〈ω〉) =1

2

h

1 −m2p −m2

π

mp(mp + 2〈ω〉)i

=

0.15 〈ω〉 = 180 MeV

0.20 〈ω〉 = 300 MeV , Ep = 2E0

7

τ : decay time of energy Ep > E0 during journey through the Universe

1

τ(Ep)= − kT

2π2γ2

Z ∞

ω0

dω σ(ω)K(ω) ω lnh

1 − e−ω/(2γkT )

i

| {z }from Planck distribution

(F.W. Stecker, ’68)

Cross-section σ is well-known from experiments:

proton at rest in γ radiation σ ≈ 0.1 mb (e.g. K.K. Wilson, ’58)

⇒ Computation of τ and corresp. path length ℓ ≃ τc ∼ 10 . . . 20 Mpc

Various corrections (redshift for protons, additional photopion productions channels etc.)

still decrease τ and ℓ a little.

Also for protons: very high starting energy → energy loss more rapid

⇒ Range with E > GZK cutoff is maximally ∼ 50 . . . 100 Mpc

Heavy nuclei: photodisintegration → attenuation length even shorter

8

Auger Project: GZK

Energy attenuation of protons

Protons: photopion threshold @ ~50 EeV

source

Photons: pair production threshold @ ~200 TeV

Nuclei: photodisintegration above 50 EeV

Neutrinos: no problem!

For E>100 EeV, the source must be within ~50 Mpc

320 EeV

20 50

Energy at

9

[Rgalactic plane ∼ 15 kpc] ≪ attenuation length ≪ [Rvisible Universe ∼ 14 Gpc]

Source should be near-by (e.g. Virgo galaxy cluster, 20 Mpc).

Homogeneously distributed sources → pile-up at E<∼E0

(5 · 1017 eV < E < E0 : p+ γ → p + e+ + e− but ∆E is small).

No sufficient acceleration mechanism is known, in particular not in our vicinity

→ Exceeding the GZK cutoff would be mysterious

Observations

1963 J. Linsley et al. (New Mexico) one event at 1020 eV

1971 K. Suga et al. (Tokyo) new super-GZK event

1991 Fly’s Eye (Utah) claims world record: 3 · 1020 eV (= 48 J)

21st century : AGASA (Japan) numerous super-GZK events.

Spectrum agrees with Yakutsk (Russia) and Haverah Park (England),

but in contradiction to HiRes (Utah): seems to confirm cutoff.

De Marco/Blasi/Olinto (2003): Discrepancy might be explained statistically.

10

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220

2

4

6

8

10

12

E19 (eV)

Num

ber

Yakutsk

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220

2

4

6

8

10

12

Haverah Park

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220

2

4

6

8

10

12

Num

ber

Num

ber

AGASA

Super-GZK events at AGASA, Haverah Park and Yakutsk

11

Flux: E > 1012 eV : ∼ 10 primary particles / (m2 min)

E > 1018.5 eV : ∼ 1 primary particle / (km2 year)

Discrepancy between methods of detection ?

• AGASA etc. detect air showers on surface of the earth

secondary: π,K · · · → µ . . . (µ survive ∼ 15 km to the earth).

O(1) particle per GeV , up to 1011 particles

→ conclusions about energy of primary particle

{ shower is reconstructed with numerical methods, “max. likelihood” }

• HiRes: Fluorescence : bluish/UV light emitted from excited Ni2in nights without moon light and clouds visible by telescopes.

Heavy nuclei as primary particles → higher shower onset type of primary particle

[ Record at 3 · 1020 eV was presumably a heavy nucleus, e.g. oxygen ]

Spallation : heavy nuclei break apart after a while (collision in gas clouds)

plus photodisintegration → high proton fraction hints at a long path.

12

13

log10(E) [eV]18.5 19 19.5 20 20.5

]-1

sr

-1 s-2

m-1

log1

0(fl

ux)

[eV

-38

-37

-36

-35

-34

-33

-32

-31

-30

log10(E) [eV]18.5 19 19.5 20 20.5

]-1

sr

-1 s-2

m-1

log1

0(fl

ux)

[eV

-38

-37

-36

-35

-34

-33

-32

-31

-30

log10(E) [eV]18.5 19 19.5 20 20.5

]-1

sr

-1 s-2

m2)

[eV

3lo

g10(

flux

E

22.5

23

23.5

24

24.5

25

25.5

log10(E) [eV]18.5 19 19.5 20 20.5

]-1

sr

-1 s-2

m2)

[eV

3lo

g10(

flux

E

22.5

23

23.5

24

24.5

25

25.5

Data from AGASA vs. HiRes

Pierre Auger Project (in Argentina, planing since 1992, operating in part since 2003)

combines both, hopes to resolve this issue

• on ground: water Cerenkov array

1600 water tanks over 3000 km2 (sizable statistics)

• Fluorescence: 24 telescopes verify correlation and energy calibration

→ E to ≈ 22% precision

14

Data by the Pierre Auger Collaboration (presented in July 2007 in Mexico)

Spectrum decays as ∝ E−2.69

Clear reduction above E ≈ 1019.6 eV, but new super-GZK events !

Nov. 2007: Analysis and interpretation of arrival directions

→ Appendix

Not really a “cutoff”, space for speculations !

E.g. violation of Lorentz symmetry (crucial for σ and K !) . . .

15

II. Lorentz Symmetry

So far assumed to hold (even at γ-factor ∼ 1011), central characteristic of relativity

• Special RT : holds globally [ H.A. Lorentz (1904), H. Poincare, A. Einstein (1905) ]

• General RT : holds locally [ A. Einstein (∼ 1915)]

Field Φ (scalar, 4-vector, tensor, spinor) transforms in a representation D of the Lorentz

group SO(1, 3),

Φ(x) → D(Λ)Φ(Λ−1x) , Λ ∈ SO(1, 3) .

In particular scalars remain Lorentz invariant (LI).

Theorem : { LI and Locality } ⇒ CPT Invariance

W. Pauli, G. Luders, R. Jost (1957)

Discrete sym. under simultaneous

charge conjugation (C), space reflection (P) and time inversion (T) .

16

O.W. Greenberg (2002): CPT violation ⇒ LI violation (LIV) (not ⇐)

[ CPT tests with K0 vs. K0 : ∆ mass (relative) < 8 · 10−19 ]

Direct tests of Lorentz sym. through invariance of c :

• Michelson-Morley type : |∆c/c|<∼ 10−11

• Atomic physics : precision tests of specific LIV parameters

e.g. spin coupling of p , n, e− to a possible “tensor background field” :

for some parameters : rel. deviation < 10−27

• Outlook: atomic clocks on ISS etc.

Impressive, but: CPT conserving LIV ∝ E2

assume e.g. on Planck scale (MPlanck = 1/√G ≈ 1028 eV) LIV ∼ O(1)

accelerators E < 1013 eV → LIV ∼ O(10−30)

on the other hand:

Laboratory LIV ∼ O(10−25), CPT violation ∝ E → at MPlanck : LIV ∼ O(10−10)

→ CPT even terms are more interesting.

17

Cosmic rays: unique hope for measuring effects not far from MPlanck ; long path !

Tests for theories like string, (loop-)quantum gravity etc. are conceivable.

They like to install new fields in the vacuum, which may yield LIV.

Systematic Approach: (A. Kostelecky et al., since 1998)

“Standard Model Extension” , Lorentz sym. breaks spontaneously. Example:

L = iψγµ∂µψ − gψφψ − ig′Gµνψγ

µ∂νψ + . . .

φ : Higgs field, SM: m = g〈φ0〉analogous: tensor field 〈G00〉 > 0 , otherwise 〈Gµν〉 = 0 .

⇒ modified dispersion relation for each type of particle, depending on its coupling to Gµν

Kostelecky: > 100 parameters of this kind

preserve “all usual properties of the SM” (e.g. E, ~p ) except for LI (and CPT).

Special RT : Goldstone boson , photon

General RT : various scenarios

Problem: why LIV at high energy ?

18

Pragmatic Approach (S. Coleman / S. Glashow ’99)

Leff with explicit LIV parameters of mass dim. < 4 (renormalisable),

CPT and gauge invariance persist, in addition SO(3) sym. in a “preferred frame”

• Boson field ~Φ :

L = · · · + 1

2

3X

i=1

∂iφaεab∂

iφb

(ε : sym.)

• Dirac spinor :L = · · · + iψ~γ~∂ [ ε+(1 + γ5) + ε−(1 − γ5) ]ψ

• Pure gauge terms, e.g. for U(1) : Ei = F 0i , Bi = 12ǫijkFjk

rot’sym., ren’able terms : ~E2 − ~B

2, ~E · ~B

| {z }LI

, ~B2 , ~A · ~B| {z }breaks CPT

→ use also in YM theories :P

a~Ba · ~Ba (a : generators)

Leads to quasi-SM with 46 LIV parameters (many from fermion generation mixing)

with gauge anomaly = 0 (gauge invariance on quantum level)

19

Example: real scalar field with renormalised propagator

−iD−1= (p

2 −m20)f(p

2) + ε~p

2g(p

2)

[ Minkowski space with c = 1, p2 = E2 − ~p 2, m0: renormalised at ε → 0 ]

f, g : smooth functions with normalisation f(m20) = g(m2

0) = 1

LIV perturbation in O(ε) shifts the poles to

E2

= ~p2+m

20 − ε~p

2= ~p

2c2P +m

2c4P

with m =m0

1 + ε, c2P = 1 − ε

Each particle receives its own Maximal Attainable Velocity (MAV).

[ Group velocity : ∂E∂|~p| = |~p|

q

|~p|2+m2c2P

cP ]

Correction becomes significant when ε~p 2/m20 ∼ O(1)

⇒ tiny ε could be manifest at some tremendous energy ! (new hierarchy problem)

20

Applications:

• Decay at ultra high energy : particle0 → P

a particlea(m negligible)

Decay condition:

c0|~p0| =X

a

ca|~pa| ≥ cmin

X

a

|~pa| ≥ cmin|~p0|

⇒ c0 ≥ cmin :=mina ca

• Charged particle with cP/cγ = 1 + ε > 1 :

“Vacuum Cerenkov radiation” at v > cγ ,

i.e. E > m/q

1 − c2γ/c2P ≃ m/

√2ε

◮ Protons survive E ≃ 1020eV ⇒ εp <m2p

2E2 ≈ 5 · 10−23

better than bound from atomic physics (but only upper bound)

◮ Cosmic e± observed up to E ≃ 1 TeV ⇒ εe < 10−13

21

• GZK Cutoff

Consider head-on collision p+γ → ∆(1232) with cγ = c∆ = 1 , cp = 1 − ε

Constraint for a ∆ resonance :

m2∆ < (E + ω)2 − (pi − ω)2 ≃ E2 − p2

i| {z }+2ω(E + pi)

E, pi for a proton in the “laboratory” :

E2 − p2i (1 − ε)2 = m2

p(1 − ε)4 −→| {z }E≫mp , |ε|≪1

E2 − p2i| {z }≃ m2

p − 2 εE2

⇒ m2∆ −m

2p + 2 εE

2< 4ωE

22

E

4 ω E

m∆2 − m p

2

m∆2 − m p

2 + 2 ε E2

m2∆ −m

2p + 2 εE

2< 4ωE

◮ At ε = 0 : minimal energy E0 =m2

∆−m2p

4ω

◮ With ε included, only soluble if

ε <ω

2E0

≃ 2ω2

m2∆ −m2

p

|ω=6·10−4 eV = 1.9 · 10−25

A tiny ε could remove the GZK cutoff !

[ For slow protons the resonance p+ γ → ∆ persists.]

23

This rules out the ∆ channel for the photopion production.

Next candidate : p + γ → p∗(1435) → p + π

at ultra high energy : decay only for cπ − cp < 5 · 10−24

we could close this channel too . . .

Farrar/Biermann (1998) :

the 5 top events ( > 1020 eV) all originate from the direction of a quasar.

[ Quasi-stellar radio source: extremely bright centre of a young galaxy ]

Coleman/Glashow : primary particle of the super-GZK events could be neutrons :

• cn < cp : no β-decay at high energy

• cn < c∆ : protected from the GZK cutoff

• hardly deflected by magnetic fields

[ Today (with O(100) super-GZK events) quasar hypothesis out of fashion,

but clustering of directions revitalised, neutral primary particles (?)]

24

Maximal Attainable Velocities of the Neutrinos

Three bases for the neutrino states:

eigenstates of flavour, of mass m0 or of MAV cν.

In principle neutrino oscillation is possible even at mν = 0 , but not compatible with phen. data.

(Lipari/Lusignoli ’99)

We concentrate on the oscillation νµ ↔ ντ .

Assumption: dominant effect due to flavour-mixing of the mass states,

plus ev. sub-dominant effect from

∆v = MAV(ν1) − MAV(ν2)

θv = mixing angle of |νµ〉 and |ντ〉 in MAV basis .

∆v and θv modify the life time of νµ .

25

Example of the MACRO Collaboration (Gran Sasso) :

Neutrino energy (GeV)10 210

Su

rviv

al p

rob

abili

ty0

0.2

0.4

0.6

0.8

1

1± = vθ, sin 2 -25 v = 2 10∆

Survival probability of νµ over 10 000 km at ∆v = 2 · 10−25 , sin 2θv = 0 , 1 , −1 .

Sensitivity at high energy of the νµ.

Consider a cosmic ν with O(100) GeV , for mν <∼ 1 eV : γ >∼ 1011 (like proton).

Detection of upward directed µ from νµ +N → µ+ . . .

multi-Coulomb scattering → reconstruction of Eµ and Eνµ ,

58 events with Eνµ > 130 GeV, compare to flux at low Eνµ

26

Result (2004) :

-1 -0.5 0 0.5 110

-26

10-25

10-24

10-23 Excluded

sin 2θv

∆v/2

Variation of ∆v and θv does not improve the fit.

For arbitrary θv : |∆v| < 6 · 10−24 (90 % C.L.)

[ Agreement with Super-Kamiokande K2K data (Fogli et al. ’99)]

27

III. Cosmic γ-rays

We now consider the photons themselves (so far in the background).

Highest energy Eγ > 50 TeV from Crab nebula (rest of a supernova, distance : 2 kpc).

Strongest sources beyond our galaxy :

Blazars, e.g. Markarian 501 (HEGRA, 1999), Eγ ≈ 20 TeV,

distance 157 Mpc (from redshift).

Subset of “Active Galactic Nuclei”, environment of a super-massive Black Hole,

driven by swallowed matter → emits γ, e± . . .A few hundreds are known, here distance and direction can often be determined.

New puzzle similar to GZK

We expect pair creation with IR background photons

γUV(E) + γIR(ω) → e+

+ e−

In centre-of-mass system: ω = E/γ = γω → condition: ω2 = Eω > m2e .

28

Example: for E ∼ 10 TeV scattering at ω>∼ 3 meV.

Despite the low density, this cross-section σ is sufficient,

to practically exclude Eγ over such long distances.

Stecker/Glashow ’01 : Way out analogous to GZK

ce = cγ + ε

Condition for head-on collisions : 2Eω − E2ε > 2m2e

ε > 0 could increase the energy threshold, or avoid pair creation completely

⇒ Universe becomes transparent for all photons.

No pair creation for ε ≥ 2E2(Eω −m2

e)|E=20 TeV, ω=0.003 eV = 2 · 10−15

below bound for vacuum Cerenkov radiation of the electron, ε < 10−13 .

29

γ-Ray-Bursts (GRB)

Emitted in powerful energy eruptions for short periods (sec. to min.),

temporarily brightest γ source in the sky.

Sources are small, merger of neutron stars or Black Holes or . . .

Known since 1973, homogeneous distribution, Eγ = 104 . . . 108 eV

Discovery from satellites, redshift measured from ground.

Direct identification from ground more efficient (La Silla, Chile).

2005: Observation from 4 Gpc , i.e. from Early Universe ( < 109 y ).

Amelino-Camelia et al. : test for dispersion relation: vγ(E) = const. ?

30

Simplest attempt : mγ > 0 , vγ = ∂E∂p 6= const.

ener

gy E

pγ

mγ > 0mγ = 0

Almost same time of arrival after a long journey → mγ < 10−6 eV

However :

much better bound from laboratory mγ < 6 · 10−17 eV (CERN Data Booklet)

We stay with mγ = 0 .

Here Coleman/Glashow ansatz E2 = p2c2γ does not help.

31

• “Doubly Special Relativity”:

Class of theoretical approaches, which try to introduce a second absolute bound,

in addition to c (Galilei: 0, Einstein: 1, are there more ?)

Example: H.S. Snyder (1947): absolute minimal length

(maybe Planck length 1/MPlanck ≃ 10−35 m)

Idea: proceed as with angular momentum operator L3 from a 5d perspective (c = 1).

S = x20 − x

21 − x

22 − x

23 − x

2

4

S = a2 : 4d de Sitter space with inside 5d light cone.

Generation of transformations, which leave S invariant:

L3 =~

i(x1∂2 − x2∂1) invariant x

21 + x

22, x0, x3, x4

X =a

i(x1∂4 − x4∂1) . . . x

21 + x

24, . . . (4d LIV)

T = ai(x0∂4 + x4∂0) . . . x20 − x

24, . . . (4d LIV)

32

Spectrum of X is discrete

Xψ = λψ ,

„x1

x4

«

= r

„sinϕcosϕ

«

, X = ai∂ϕ , ψ ∝ exp( iaϕλ)

ψ(ϕ) = ψ(ϕ+ 2π) ⇒ λ = na , n ∈ ZZ

Position operators do not commute: [X,Y ] = ia~L3 etc.

→ new uncertainty relation min(∆X ∆Y ) ∝ a2

Minimal length a as an absolute constant, 4d non-locality, but 5d LI.

Interpretation as event horizon in a mini Black Hole

matches a = Planck length (Doplicher/Fredenhagen/Roberts ’95)

Currently popular version: commutators as constant “tensor field”

[Xµ, Xν] = iΘµν

observer independent, sets min. area (tensor under deformed Lorentz trafo).

33

Non-commutativity affects pure U(1) gauge field :

picks up a YM-type self-coupling → deformed photon dispersion

1-loop result takes the form (Matusis/Susskind/Toumbas ’00)

E2 = ~p 2 +C

(pΘ)2

[ on quantum level the new UV term Θµν causes also IR divergence (additional uncertainty !) ]

◮ Amelino-Camilina et al. (2003) :

Analysis of GRB radiation ⇒ ‖Θ‖ > 10−40 cm2 , otherwise effect should be larger.

However:

• Θ = 0 is not excluded in this way.

• C < 0 i.e. 1-loop result is actually IR unstable ! (Landsteiner/Lopez/Tytgat ’01).

(SUSY cancels IR divergence . . . )

34

NC QED revisited non-perturbatively (W.B./Nishimura/Susaki/Volkholz ’06)

• Commutative plane (x3, x4) → Lattice

includes Euclidean time (enables transition to Minkowski signature)

• NC plane (x1, x2), [x1, x2] = iθ

Lattice structure : exp“

i2πa xµ”

= 11 (µ = 1, 2)

Momenta commute, usual periodicity

eikµxµ = e

i(kµ+2πa )xµ

11 = ei(kµ+2πa )xµ e−ikνxν = · · · = 11 exp

“iπ

aθ(k2 − k1)

”

⇒ θ

2akµ ∈ ZZ : momenta discrete, lattice periodic

Periodic N ×N lattice: kµ = 2πaNnµ (nµ ∈ ZZ) ⇒ θ = 1

πNa2

35

Double Scaling Limit :continuum a → 0

infinite volume Na → ∞

ff

Na2 = const.

Simultaneous UV and IR limit, which keeps θ = const. (Szabo ’01)

U(1) gauge theory on a NC lattice can be mapped onto a

“twisted Eguchi-Kawai model” (U(N) matrices in one point) (Ambjørn et al. ’01)

→ numerically tractable

Yang-Mills type self-interaction and gauge transformations are non-local on scalep

|θ|.

In this range: gauge invariant open Wilson lines

carry momentum → order parameters for spont. breaking of transl. sym.

Numerical observation:

Double Scaling Limit β ≡ 1g2

∝√N stabilises a variety of observables (a = 1/β)

36

Phase diagram : weak ↔ moderate ↔ strong coupling

Ph

as

e

We

ak

Co

up

li n

g

S t r o n g C o u p l i n g P h a s e70605040302010

6

5

4

3

2

1

0

β=

1 /

g

N

DSL θ = 14

2

B r

o k

e n

P

h a

s e

DSL θ = 1.5

Double Scaling Limit β ∝√N always leads to the phase of broken symmetry.

That phase could describe a stable cont. limit for the NC photon.

37

Dispersion relation: determined from exp. decay in comm. plane E(p = p3)|p1=p2=0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

E

p

N=35,β=2.00N=25,β=1.69N=15,β=1.29

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

E

p

N=25,β=0.92N=35,β=1.20N=45,β=1.50N=55,β=1.74N=65,β=2.00

sym. phase broken phase

consistent with neg. IR divergence IR stable

“tachyonic′′ behaviour Goldstone boson

Photon may survive in an NC world,

but explicit prediction for the deformed dispersion relation is outstanding.

38

• Return to a pragmatic ansatz: J. Ellis et al. (2006/7)

c2|~p|2 = E

2“

1 +E

M

”

→ vγ(E) =∂E

∂|~p| ≃ c“

1 − E

M

”

M : very heavy mass, emerges somehow from “quantum gravity foam”,

noticeable at high energy, or after a long path.

Analysis of 35 GRB’s

Data from 3 satellites [ e.g. HETE: dt = 64 ms , 4 energy channels].

High energy γ’s arrive earlier. Ansatz for the observed delay without LIV

∆tobs = dsource(1 + z)

dsource : possible delay already in the emission

z : redshift

39

With LIV: ∆tobs

1 + z= dsource +

∆E

M

1

H0| {z }aLIV

K(z) (K : complicated correction)

Enhance error bars until fits match: 1 σ evidence for LIV [ 68 %, 95 % ]

−0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

dsource

a LIV

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.15 0.2 0.25 0.3 0.35 0.4

K

∆t/(1

+z)

HETE

BATSE

SWIFT

Cautious conclusion : |M| > 1.4 · 1025 eV ≈ 0.001 MPlanck (with 95 % C.L.)

Studies of single GRBs or blazar flares (e.g. Mkn501) even conclude |M | > 0.01MPlanck

40

Conclusions:

Cosmic rays: unique opportunity for phenomenological access to tremendous energies.

In the centre-of-mass frame, relevant processes are harmless low energy events

→ Question of LI is crucial !

GZK and γ-TeV Puzzle :

Why is the Universe surprisingly transparent for

• protons with Ep>∼ 1020 eV • photons with Eγ >∼ 10 TeV ?

Open question; LIV provides a class of proposals for a solution.

LIV has not been detected anywhere — we discussed failed attempts with cosmic neutrinos

and GRBs. But established LI precision does not exclude proposed solutions.

New projects include : Telescope Array (TA), Extreme Universe Space Observatory (EUSO),

Orbiting Wide-angle Light-collectors (OWL): search for fluorescence light from satellites

Pierre Auger plant completed only in summer 2008 . . .

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News in Nov. 2007 : AGN Hypothesis

[Active Galactic Nuclei: in centre a super-massive black hole (> 106 solar masses), nucleus

attracts and absorbs large quantities of matter, but emits high-E particles (mechanism ?)]

Pierre Auger Collab. analyses UHECRs detected from Jan. 2004 - May 2006

Hypothesis: directions are clustered and correlated with locations of nearby AGN

3 parameters: ψ : angular range around UHECR direction

Emin : threshold for UHECR

Rmax : max. distance to “nearby” AGN (from redshift)

tuning → (ψ,Emin, Rmax) = (3.10, 5.6 · 1019 eV, 75 Mpc)

yields max. correlation, captures 12 our of 15 UHECR

(for isotropic sources: 3.2 expected [at fixed parameters . . . ])

Rmax short, ≈ straight UHECR propagation conceivable

Check with data from May 2006 - Aug. 2007: captures 8 out of 13 UHECRs (2.8 expected)

42

Celestial sphere with circles of radius 3.1◦ at arrival directions of 27 UHECRs detected by

the Pierre Auger Observatory. Asterisks: 472 AGN with R < 75 Mpc.

Dashed : supergalactic plane; white : Centaurus A

For clustering PA Collab. claims 99 % C.L., correlation less clear

43

Critics :

• Variation of (ψ,Emin, Rmax) is discussed only vaguely.

• Statistics still small, world data before essentially isotropic

[AGASA (1996): slight signal for clustering, contradicted by HiRes]

Consistent with world data ?

• Gorbunov/Tinyakov/Tkachev/Troisky :

Flux ∝ 1/R2 → nearest AGN should be dominant sources

in particular Cen A and Virgo should contribute each ≈ 6 events out of 27

Cen A in business, but Virgo delivers none

⇒ AGN Hypothesis disfavoured at 99 % C.L.

[However:

argument could be evaded if AGN are episodic UHECR sources]

• D. Fargion: short Rmax favours heavy nuclei as primaries

• Hypothesis supported by new Irkutsk data analysis, but not by HiRes; AGASA pending.

44